Properties

Label 4009.2.a.e.1.6
Level $4009$
Weight $2$
Character 4009.1
Self dual yes
Analytic conductor $32.012$
Analytic rank $0$
Dimension $82$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4009,2,Mod(1,4009)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4009, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4009.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4009 = 19 \cdot 211 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4009.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(32.0120261703\)
Analytic rank: \(0\)
Dimension: \(82\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Character \(\chi\) \(=\) 4009.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.39399 q^{2} +1.52449 q^{3} +3.73121 q^{4} -2.51919 q^{5} -3.64962 q^{6} -1.00997 q^{7} -4.14450 q^{8} -0.675931 q^{9} +O(q^{10})\) \(q-2.39399 q^{2} +1.52449 q^{3} +3.73121 q^{4} -2.51919 q^{5} -3.64962 q^{6} -1.00997 q^{7} -4.14450 q^{8} -0.675931 q^{9} +6.03093 q^{10} +1.01663 q^{11} +5.68819 q^{12} -6.16777 q^{13} +2.41786 q^{14} -3.84048 q^{15} +2.45950 q^{16} -0.961955 q^{17} +1.61818 q^{18} +1.00000 q^{19} -9.39963 q^{20} -1.53968 q^{21} -2.43379 q^{22} -4.71882 q^{23} -6.31825 q^{24} +1.34633 q^{25} +14.7656 q^{26} -5.60392 q^{27} -3.76840 q^{28} +0.466130 q^{29} +9.19409 q^{30} -7.02934 q^{31} +2.40098 q^{32} +1.54983 q^{33} +2.30292 q^{34} +2.54430 q^{35} -2.52204 q^{36} +5.24361 q^{37} -2.39399 q^{38} -9.40270 q^{39} +10.4408 q^{40} -8.70221 q^{41} +3.68600 q^{42} +5.38819 q^{43} +3.79324 q^{44} +1.70280 q^{45} +11.2968 q^{46} +6.30953 q^{47} +3.74948 q^{48} -5.97997 q^{49} -3.22310 q^{50} -1.46649 q^{51} -23.0132 q^{52} +5.52900 q^{53} +13.4157 q^{54} -2.56107 q^{55} +4.18581 q^{56} +1.52449 q^{57} -1.11591 q^{58} +4.06983 q^{59} -14.3296 q^{60} +0.502132 q^{61} +16.8282 q^{62} +0.682668 q^{63} -10.6669 q^{64} +15.5378 q^{65} -3.71029 q^{66} -4.59271 q^{67} -3.58925 q^{68} -7.19380 q^{69} -6.09104 q^{70} +13.8847 q^{71} +2.80140 q^{72} -10.1040 q^{73} -12.5532 q^{74} +2.05246 q^{75} +3.73121 q^{76} -1.02676 q^{77} +22.5100 q^{78} -5.90648 q^{79} -6.19594 q^{80} -6.51532 q^{81} +20.8330 q^{82} -9.85377 q^{83} -5.74488 q^{84} +2.42335 q^{85} -12.8993 q^{86} +0.710610 q^{87} -4.21340 q^{88} +4.69318 q^{89} -4.07649 q^{90} +6.22924 q^{91} -17.6069 q^{92} -10.7162 q^{93} -15.1050 q^{94} -2.51919 q^{95} +3.66028 q^{96} -6.31958 q^{97} +14.3160 q^{98} -0.687169 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9} + 4 q^{10} + 41 q^{11} + 26 q^{12} + 13 q^{13} + 22 q^{14} + 41 q^{15} + 87 q^{16} + 12 q^{17} + 24 q^{18} + 82 q^{19} + 26 q^{20} + 29 q^{21} + 2 q^{22} + 59 q^{23} + 16 q^{24} + 67 q^{25} + 24 q^{26} + 42 q^{27} - 2 q^{28} + 101 q^{29} - 22 q^{30} + 48 q^{31} + 69 q^{32} + 3 q^{33} + q^{34} + 38 q^{35} + 82 q^{36} + 16 q^{37} + 15 q^{38} + 82 q^{39} + 20 q^{40} + 86 q^{41} - q^{42} + 9 q^{43} + 82 q^{44} - 8 q^{45} + 43 q^{46} + 24 q^{47} + 34 q^{48} + 76 q^{49} + 82 q^{50} + 57 q^{51} - 22 q^{52} + 39 q^{53} + 17 q^{54} - 21 q^{55} + 50 q^{56} + 12 q^{57} + 33 q^{58} + 79 q^{59} + 87 q^{60} + 4 q^{61} + 40 q^{62} + 44 q^{63} + 90 q^{64} + 66 q^{65} - 39 q^{66} + 33 q^{67} - 9 q^{68} + 60 q^{69} + 30 q^{70} + 168 q^{71} + 15 q^{72} - 28 q^{73} + 35 q^{74} + 55 q^{75} + 89 q^{76} + 19 q^{77} - 41 q^{78} + 121 q^{79} + 64 q^{80} + 110 q^{81} + 41 q^{82} + 28 q^{84} + 17 q^{85} + 80 q^{86} + 29 q^{87} + 49 q^{88} + 83 q^{89} - 42 q^{90} + 38 q^{91} + 71 q^{92} - q^{93} + 89 q^{94} + 9 q^{95} + 35 q^{96} - 23 q^{97} + 135 q^{98} + 93 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.39399 −1.69281 −0.846405 0.532540i \(-0.821238\pi\)
−0.846405 + 0.532540i \(0.821238\pi\)
\(3\) 1.52449 0.880165 0.440082 0.897957i \(-0.354949\pi\)
0.440082 + 0.897957i \(0.354949\pi\)
\(4\) 3.73121 1.86560
\(5\) −2.51919 −1.12662 −0.563308 0.826247i \(-0.690472\pi\)
−0.563308 + 0.826247i \(0.690472\pi\)
\(6\) −3.64962 −1.48995
\(7\) −1.00997 −0.381732 −0.190866 0.981616i \(-0.561130\pi\)
−0.190866 + 0.981616i \(0.561130\pi\)
\(8\) −4.14450 −1.46530
\(9\) −0.675931 −0.225310
\(10\) 6.03093 1.90715
\(11\) 1.01663 0.306524 0.153262 0.988186i \(-0.451022\pi\)
0.153262 + 0.988186i \(0.451022\pi\)
\(12\) 5.68819 1.64204
\(13\) −6.16777 −1.71063 −0.855315 0.518108i \(-0.826637\pi\)
−0.855315 + 0.518108i \(0.826637\pi\)
\(14\) 2.41786 0.646199
\(15\) −3.84048 −0.991608
\(16\) 2.45950 0.614874
\(17\) −0.961955 −0.233308 −0.116654 0.993173i \(-0.537217\pi\)
−0.116654 + 0.993173i \(0.537217\pi\)
\(18\) 1.61818 0.381408
\(19\) 1.00000 0.229416
\(20\) −9.39963 −2.10182
\(21\) −1.53968 −0.335987
\(22\) −2.43379 −0.518887
\(23\) −4.71882 −0.983943 −0.491971 0.870611i \(-0.663724\pi\)
−0.491971 + 0.870611i \(0.663724\pi\)
\(24\) −6.31825 −1.28971
\(25\) 1.34633 0.269265
\(26\) 14.7656 2.89577
\(27\) −5.60392 −1.07847
\(28\) −3.76840 −0.712160
\(29\) 0.466130 0.0865582 0.0432791 0.999063i \(-0.486220\pi\)
0.0432791 + 0.999063i \(0.486220\pi\)
\(30\) 9.19409 1.67860
\(31\) −7.02934 −1.26251 −0.631253 0.775577i \(-0.717459\pi\)
−0.631253 + 0.775577i \(0.717459\pi\)
\(32\) 2.40098 0.424438
\(33\) 1.54983 0.269792
\(34\) 2.30292 0.394947
\(35\) 2.54430 0.430065
\(36\) −2.52204 −0.420340
\(37\) 5.24361 0.862044 0.431022 0.902341i \(-0.358153\pi\)
0.431022 + 0.902341i \(0.358153\pi\)
\(38\) −2.39399 −0.388357
\(39\) −9.40270 −1.50564
\(40\) 10.4408 1.65083
\(41\) −8.70221 −1.35906 −0.679528 0.733649i \(-0.737816\pi\)
−0.679528 + 0.733649i \(0.737816\pi\)
\(42\) 3.68600 0.568761
\(43\) 5.38819 0.821691 0.410845 0.911705i \(-0.365234\pi\)
0.410845 + 0.911705i \(0.365234\pi\)
\(44\) 3.79324 0.571852
\(45\) 1.70280 0.253838
\(46\) 11.2968 1.66563
\(47\) 6.30953 0.920339 0.460170 0.887831i \(-0.347789\pi\)
0.460170 + 0.887831i \(0.347789\pi\)
\(48\) 3.74948 0.541190
\(49\) −5.97997 −0.854281
\(50\) −3.22310 −0.455815
\(51\) −1.46649 −0.205350
\(52\) −23.0132 −3.19136
\(53\) 5.52900 0.759467 0.379733 0.925096i \(-0.376016\pi\)
0.379733 + 0.925096i \(0.376016\pi\)
\(54\) 13.4157 1.82565
\(55\) −2.56107 −0.345335
\(56\) 4.18581 0.559352
\(57\) 1.52449 0.201924
\(58\) −1.11591 −0.146526
\(59\) 4.06983 0.529847 0.264923 0.964269i \(-0.414653\pi\)
0.264923 + 0.964269i \(0.414653\pi\)
\(60\) −14.3296 −1.84995
\(61\) 0.502132 0.0642914 0.0321457 0.999483i \(-0.489766\pi\)
0.0321457 + 0.999483i \(0.489766\pi\)
\(62\) 16.8282 2.13718
\(63\) 0.682668 0.0860081
\(64\) −10.6669 −1.33337
\(65\) 15.5378 1.92723
\(66\) −3.71029 −0.456706
\(67\) −4.59271 −0.561088 −0.280544 0.959841i \(-0.590515\pi\)
−0.280544 + 0.959841i \(0.590515\pi\)
\(68\) −3.58925 −0.435261
\(69\) −7.19380 −0.866031
\(70\) −6.09104 −0.728019
\(71\) 13.8847 1.64781 0.823905 0.566728i \(-0.191791\pi\)
0.823905 + 0.566728i \(0.191791\pi\)
\(72\) 2.80140 0.330148
\(73\) −10.1040 −1.18258 −0.591292 0.806457i \(-0.701382\pi\)
−0.591292 + 0.806457i \(0.701382\pi\)
\(74\) −12.5532 −1.45928
\(75\) 2.05246 0.236998
\(76\) 3.73121 0.427999
\(77\) −1.02676 −0.117010
\(78\) 22.5100 2.54876
\(79\) −5.90648 −0.664530 −0.332265 0.943186i \(-0.607813\pi\)
−0.332265 + 0.943186i \(0.607813\pi\)
\(80\) −6.19594 −0.692727
\(81\) −6.51532 −0.723925
\(82\) 20.8330 2.30062
\(83\) −9.85377 −1.08159 −0.540796 0.841154i \(-0.681877\pi\)
−0.540796 + 0.841154i \(0.681877\pi\)
\(84\) −5.74488 −0.626818
\(85\) 2.42335 0.262849
\(86\) −12.8993 −1.39097
\(87\) 0.710610 0.0761854
\(88\) −4.21340 −0.449150
\(89\) 4.69318 0.497476 0.248738 0.968571i \(-0.419984\pi\)
0.248738 + 0.968571i \(0.419984\pi\)
\(90\) −4.07649 −0.429700
\(91\) 6.22924 0.653002
\(92\) −17.6069 −1.83565
\(93\) −10.7162 −1.11121
\(94\) −15.1050 −1.55796
\(95\) −2.51919 −0.258464
\(96\) 3.66028 0.373575
\(97\) −6.31958 −0.641656 −0.320828 0.947137i \(-0.603961\pi\)
−0.320828 + 0.947137i \(0.603961\pi\)
\(98\) 14.3160 1.44613
\(99\) −0.687169 −0.0690630
\(100\) 5.02342 0.502342
\(101\) 15.0638 1.49890 0.749451 0.662060i \(-0.230318\pi\)
0.749451 + 0.662060i \(0.230318\pi\)
\(102\) 3.51077 0.347618
\(103\) −3.00494 −0.296085 −0.148043 0.988981i \(-0.547297\pi\)
−0.148043 + 0.988981i \(0.547297\pi\)
\(104\) 25.5623 2.50659
\(105\) 3.87876 0.378528
\(106\) −13.2364 −1.28563
\(107\) 8.90549 0.860927 0.430463 0.902608i \(-0.358350\pi\)
0.430463 + 0.902608i \(0.358350\pi\)
\(108\) −20.9094 −2.01201
\(109\) 9.07859 0.869571 0.434786 0.900534i \(-0.356824\pi\)
0.434786 + 0.900534i \(0.356824\pi\)
\(110\) 6.13120 0.584587
\(111\) 7.99383 0.758741
\(112\) −2.48401 −0.234717
\(113\) 0.245585 0.0231027 0.0115513 0.999933i \(-0.496323\pi\)
0.0115513 + 0.999933i \(0.496323\pi\)
\(114\) −3.64962 −0.341818
\(115\) 11.8876 1.10853
\(116\) 1.73923 0.161483
\(117\) 4.16899 0.385423
\(118\) −9.74315 −0.896930
\(119\) 0.971543 0.0890612
\(120\) 15.9169 1.45301
\(121\) −9.96647 −0.906043
\(122\) −1.20210 −0.108833
\(123\) −13.2664 −1.19619
\(124\) −26.2279 −2.35534
\(125\) 9.20430 0.823258
\(126\) −1.63430 −0.145595
\(127\) 3.74558 0.332366 0.166183 0.986095i \(-0.446856\pi\)
0.166183 + 0.986095i \(0.446856\pi\)
\(128\) 20.7346 1.83270
\(129\) 8.21423 0.723223
\(130\) −37.1974 −3.26243
\(131\) 16.8844 1.47520 0.737599 0.675239i \(-0.235960\pi\)
0.737599 + 0.675239i \(0.235960\pi\)
\(132\) 5.78275 0.503324
\(133\) −1.00997 −0.0875753
\(134\) 10.9949 0.949816
\(135\) 14.1173 1.21503
\(136\) 3.98682 0.341867
\(137\) −21.2002 −1.81126 −0.905629 0.424070i \(-0.860601\pi\)
−0.905629 + 0.424070i \(0.860601\pi\)
\(138\) 17.2219 1.46603
\(139\) 8.61031 0.730317 0.365159 0.930945i \(-0.381015\pi\)
0.365159 + 0.930945i \(0.381015\pi\)
\(140\) 9.49331 0.802332
\(141\) 9.61881 0.810050
\(142\) −33.2399 −2.78943
\(143\) −6.27031 −0.524349
\(144\) −1.66245 −0.138537
\(145\) −1.17427 −0.0975179
\(146\) 24.1889 2.00189
\(147\) −9.11640 −0.751908
\(148\) 19.5650 1.60823
\(149\) −3.13227 −0.256606 −0.128303 0.991735i \(-0.540953\pi\)
−0.128303 + 0.991735i \(0.540953\pi\)
\(150\) −4.91358 −0.401192
\(151\) −12.1668 −0.990119 −0.495060 0.868859i \(-0.664854\pi\)
−0.495060 + 0.868859i \(0.664854\pi\)
\(152\) −4.14450 −0.336163
\(153\) 0.650215 0.0525668
\(154\) 2.45805 0.198076
\(155\) 17.7082 1.42236
\(156\) −35.0834 −2.80892
\(157\) 15.3628 1.22608 0.613041 0.790051i \(-0.289946\pi\)
0.613041 + 0.790051i \(0.289946\pi\)
\(158\) 14.1401 1.12492
\(159\) 8.42890 0.668456
\(160\) −6.04854 −0.478179
\(161\) 4.76586 0.375602
\(162\) 15.5976 1.22547
\(163\) 8.17768 0.640525 0.320263 0.947329i \(-0.396229\pi\)
0.320263 + 0.947329i \(0.396229\pi\)
\(164\) −32.4698 −2.53546
\(165\) −3.90433 −0.303952
\(166\) 23.5899 1.83093
\(167\) 18.2963 1.41581 0.707903 0.706309i \(-0.249641\pi\)
0.707903 + 0.706309i \(0.249641\pi\)
\(168\) 6.38122 0.492322
\(169\) 25.0413 1.92626
\(170\) −5.80148 −0.444954
\(171\) −0.675931 −0.0516897
\(172\) 20.1044 1.53295
\(173\) −14.6066 −1.11052 −0.555258 0.831678i \(-0.687381\pi\)
−0.555258 + 0.831678i \(0.687381\pi\)
\(174\) −1.70120 −0.128967
\(175\) −1.35975 −0.102787
\(176\) 2.50039 0.188474
\(177\) 6.20441 0.466352
\(178\) −11.2354 −0.842132
\(179\) −0.802680 −0.0599951 −0.0299976 0.999550i \(-0.509550\pi\)
−0.0299976 + 0.999550i \(0.509550\pi\)
\(180\) 6.35350 0.473562
\(181\) 1.76105 0.130898 0.0654489 0.997856i \(-0.479152\pi\)
0.0654489 + 0.997856i \(0.479152\pi\)
\(182\) −14.9128 −1.10541
\(183\) 0.765494 0.0565870
\(184\) 19.5572 1.44177
\(185\) −13.2097 −0.971194
\(186\) 25.6544 1.88107
\(187\) −0.977948 −0.0715146
\(188\) 23.5422 1.71699
\(189\) 5.65977 0.411688
\(190\) 6.03093 0.437530
\(191\) −17.4284 −1.26108 −0.630539 0.776158i \(-0.717166\pi\)
−0.630539 + 0.776158i \(0.717166\pi\)
\(192\) −16.2616 −1.17358
\(193\) −14.7500 −1.06173 −0.530863 0.847458i \(-0.678132\pi\)
−0.530863 + 0.847458i \(0.678132\pi\)
\(194\) 15.1290 1.08620
\(195\) 23.6872 1.69628
\(196\) −22.3125 −1.59375
\(197\) 16.5806 1.18132 0.590659 0.806921i \(-0.298868\pi\)
0.590659 + 0.806921i \(0.298868\pi\)
\(198\) 1.64508 0.116911
\(199\) 24.9698 1.77006 0.885032 0.465531i \(-0.154137\pi\)
0.885032 + 0.465531i \(0.154137\pi\)
\(200\) −5.57985 −0.394555
\(201\) −7.00153 −0.493850
\(202\) −36.0626 −2.53735
\(203\) −0.470776 −0.0330420
\(204\) −5.47178 −0.383101
\(205\) 21.9225 1.53114
\(206\) 7.19380 0.501216
\(207\) 3.18960 0.221692
\(208\) −15.1696 −1.05182
\(209\) 1.01663 0.0703214
\(210\) −9.28573 −0.640776
\(211\) −1.00000 −0.0688428
\(212\) 20.6298 1.41686
\(213\) 21.1671 1.45034
\(214\) −21.3197 −1.45738
\(215\) −13.5739 −0.925730
\(216\) 23.2254 1.58029
\(217\) 7.09940 0.481939
\(218\) −21.7341 −1.47202
\(219\) −15.4035 −1.04087
\(220\) −9.55590 −0.644258
\(221\) 5.93312 0.399105
\(222\) −19.1372 −1.28440
\(223\) −14.2568 −0.954706 −0.477353 0.878712i \(-0.658404\pi\)
−0.477353 + 0.878712i \(0.658404\pi\)
\(224\) −2.42491 −0.162021
\(225\) −0.910024 −0.0606683
\(226\) −0.587929 −0.0391085
\(227\) −12.4302 −0.825022 −0.412511 0.910953i \(-0.635348\pi\)
−0.412511 + 0.910953i \(0.635348\pi\)
\(228\) 5.68819 0.376709
\(229\) 22.0750 1.45875 0.729377 0.684112i \(-0.239810\pi\)
0.729377 + 0.684112i \(0.239810\pi\)
\(230\) −28.4589 −1.87652
\(231\) −1.56528 −0.102988
\(232\) −1.93188 −0.126834
\(233\) 12.6830 0.830892 0.415446 0.909618i \(-0.363626\pi\)
0.415446 + 0.909618i \(0.363626\pi\)
\(234\) −9.98053 −0.652447
\(235\) −15.8949 −1.03687
\(236\) 15.1854 0.988484
\(237\) −9.00436 −0.584896
\(238\) −2.32587 −0.150764
\(239\) −2.20619 −0.142706 −0.0713532 0.997451i \(-0.522732\pi\)
−0.0713532 + 0.997451i \(0.522732\pi\)
\(240\) −9.44565 −0.609714
\(241\) 14.6666 0.944760 0.472380 0.881395i \(-0.343395\pi\)
0.472380 + 0.881395i \(0.343395\pi\)
\(242\) 23.8597 1.53376
\(243\) 6.87921 0.441302
\(244\) 1.87356 0.119942
\(245\) 15.0647 0.962447
\(246\) 31.7598 2.02493
\(247\) −6.16777 −0.392446
\(248\) 29.1331 1.84995
\(249\) −15.0220 −0.951979
\(250\) −22.0350 −1.39362
\(251\) 5.91049 0.373067 0.186534 0.982449i \(-0.440275\pi\)
0.186534 + 0.982449i \(0.440275\pi\)
\(252\) 2.54718 0.160457
\(253\) −4.79727 −0.301602
\(254\) −8.96689 −0.562632
\(255\) 3.69437 0.231351
\(256\) −28.3047 −1.76904
\(257\) −16.1650 −1.00835 −0.504173 0.863603i \(-0.668203\pi\)
−0.504173 + 0.863603i \(0.668203\pi\)
\(258\) −19.6648 −1.22428
\(259\) −5.29588 −0.329070
\(260\) 57.9747 3.59544
\(261\) −0.315072 −0.0195025
\(262\) −40.4212 −2.49723
\(263\) −26.0675 −1.60739 −0.803697 0.595039i \(-0.797137\pi\)
−0.803697 + 0.595039i \(0.797137\pi\)
\(264\) −6.42329 −0.395326
\(265\) −13.9286 −0.855628
\(266\) 2.41786 0.148248
\(267\) 7.15470 0.437861
\(268\) −17.1363 −1.04677
\(269\) −15.4349 −0.941085 −0.470542 0.882377i \(-0.655942\pi\)
−0.470542 + 0.882377i \(0.655942\pi\)
\(270\) −33.7968 −2.05681
\(271\) 5.02852 0.305461 0.152730 0.988268i \(-0.451193\pi\)
0.152730 + 0.988268i \(0.451193\pi\)
\(272\) −2.36592 −0.143455
\(273\) 9.49642 0.574749
\(274\) 50.7532 3.06612
\(275\) 1.36871 0.0825363
\(276\) −26.8416 −1.61567
\(277\) −5.50274 −0.330628 −0.165314 0.986241i \(-0.552864\pi\)
−0.165314 + 0.986241i \(0.552864\pi\)
\(278\) −20.6130 −1.23629
\(279\) 4.75135 0.284456
\(280\) −10.5449 −0.630176
\(281\) 30.4919 1.81899 0.909497 0.415710i \(-0.136467\pi\)
0.909497 + 0.415710i \(0.136467\pi\)
\(282\) −23.0274 −1.37126
\(283\) −29.2984 −1.74161 −0.870806 0.491627i \(-0.836402\pi\)
−0.870806 + 0.491627i \(0.836402\pi\)
\(284\) 51.8067 3.07416
\(285\) −3.84048 −0.227491
\(286\) 15.0111 0.887624
\(287\) 8.78895 0.518795
\(288\) −1.62290 −0.0956303
\(289\) −16.0746 −0.945567
\(290\) 2.81120 0.165079
\(291\) −9.63413 −0.564763
\(292\) −37.7001 −2.20623
\(293\) 31.3277 1.83018 0.915090 0.403249i \(-0.132119\pi\)
0.915090 + 0.403249i \(0.132119\pi\)
\(294\) 21.8246 1.27284
\(295\) −10.2527 −0.596934
\(296\) −21.7322 −1.26316
\(297\) −5.69709 −0.330578
\(298\) 7.49864 0.434385
\(299\) 29.1046 1.68316
\(300\) 7.65816 0.442144
\(301\) −5.44189 −0.313665
\(302\) 29.1272 1.67608
\(303\) 22.9646 1.31928
\(304\) 2.45950 0.141062
\(305\) −1.26497 −0.0724317
\(306\) −1.55661 −0.0889856
\(307\) −18.9351 −1.08069 −0.540343 0.841445i \(-0.681705\pi\)
−0.540343 + 0.841445i \(0.681705\pi\)
\(308\) −3.83105 −0.218294
\(309\) −4.58100 −0.260604
\(310\) −42.3934 −2.40779
\(311\) −24.7482 −1.40334 −0.701670 0.712502i \(-0.747562\pi\)
−0.701670 + 0.712502i \(0.747562\pi\)
\(312\) 38.9695 2.20621
\(313\) 5.02782 0.284189 0.142095 0.989853i \(-0.454616\pi\)
0.142095 + 0.989853i \(0.454616\pi\)
\(314\) −36.7784 −2.07552
\(315\) −1.71977 −0.0968982
\(316\) −22.0383 −1.23975
\(317\) −11.1787 −0.627860 −0.313930 0.949446i \(-0.601646\pi\)
−0.313930 + 0.949446i \(0.601646\pi\)
\(318\) −20.1787 −1.13157
\(319\) 0.473880 0.0265322
\(320\) 26.8720 1.50219
\(321\) 13.5763 0.757757
\(322\) −11.4094 −0.635823
\(323\) −0.961955 −0.0535246
\(324\) −24.3100 −1.35056
\(325\) −8.30383 −0.460614
\(326\) −19.5773 −1.08429
\(327\) 13.8402 0.765366
\(328\) 36.0663 1.99143
\(329\) −6.37242 −0.351323
\(330\) 9.34694 0.514532
\(331\) −23.4557 −1.28924 −0.644622 0.764501i \(-0.722985\pi\)
−0.644622 + 0.764501i \(0.722985\pi\)
\(332\) −36.7665 −2.01782
\(333\) −3.54432 −0.194228
\(334\) −43.8011 −2.39669
\(335\) 11.5699 0.632132
\(336\) −3.78685 −0.206589
\(337\) −25.5783 −1.39334 −0.696669 0.717392i \(-0.745336\pi\)
−0.696669 + 0.717392i \(0.745336\pi\)
\(338\) −59.9488 −3.26079
\(339\) 0.374392 0.0203342
\(340\) 9.04202 0.490372
\(341\) −7.14620 −0.386989
\(342\) 1.61818 0.0875009
\(343\) 13.1093 0.707838
\(344\) −22.3313 −1.20403
\(345\) 18.1226 0.975685
\(346\) 34.9680 1.87989
\(347\) −15.2290 −0.817537 −0.408768 0.912638i \(-0.634042\pi\)
−0.408768 + 0.912638i \(0.634042\pi\)
\(348\) 2.65144 0.142132
\(349\) 10.3873 0.556019 0.278010 0.960578i \(-0.410325\pi\)
0.278010 + 0.960578i \(0.410325\pi\)
\(350\) 3.25522 0.173999
\(351\) 34.5637 1.84487
\(352\) 2.44090 0.130100
\(353\) 22.6795 1.20711 0.603555 0.797321i \(-0.293750\pi\)
0.603555 + 0.797321i \(0.293750\pi\)
\(354\) −14.8533 −0.789446
\(355\) −34.9782 −1.85645
\(356\) 17.5112 0.928093
\(357\) 1.48111 0.0783885
\(358\) 1.92161 0.101560
\(359\) 26.9195 1.42076 0.710378 0.703820i \(-0.248524\pi\)
0.710378 + 0.703820i \(0.248524\pi\)
\(360\) −7.05726 −0.371950
\(361\) 1.00000 0.0526316
\(362\) −4.21594 −0.221585
\(363\) −15.1938 −0.797467
\(364\) 23.2426 1.21824
\(365\) 25.4539 1.33232
\(366\) −1.83259 −0.0957910
\(367\) −23.1032 −1.20598 −0.602988 0.797750i \(-0.706023\pi\)
−0.602988 + 0.797750i \(0.706023\pi\)
\(368\) −11.6059 −0.605001
\(369\) 5.88209 0.306210
\(370\) 31.6239 1.64405
\(371\) −5.58411 −0.289912
\(372\) −39.9842 −2.07308
\(373\) 17.6456 0.913653 0.456827 0.889556i \(-0.348986\pi\)
0.456827 + 0.889556i \(0.348986\pi\)
\(374\) 2.34120 0.121061
\(375\) 14.0319 0.724602
\(376\) −26.1498 −1.34858
\(377\) −2.87498 −0.148069
\(378\) −13.5495 −0.696909
\(379\) 3.63529 0.186732 0.0933662 0.995632i \(-0.470237\pi\)
0.0933662 + 0.995632i \(0.470237\pi\)
\(380\) −9.39963 −0.482191
\(381\) 5.71009 0.292537
\(382\) 41.7236 2.13476
\(383\) 17.0434 0.870877 0.435438 0.900219i \(-0.356593\pi\)
0.435438 + 0.900219i \(0.356593\pi\)
\(384\) 31.6097 1.61308
\(385\) 2.58660 0.131825
\(386\) 35.3113 1.79730
\(387\) −3.64204 −0.185135
\(388\) −23.5797 −1.19708
\(389\) 8.92973 0.452755 0.226378 0.974040i \(-0.427312\pi\)
0.226378 + 0.974040i \(0.427312\pi\)
\(390\) −56.7070 −2.87147
\(391\) 4.53930 0.229562
\(392\) 24.7840 1.25178
\(393\) 25.7401 1.29842
\(394\) −39.6938 −1.99975
\(395\) 14.8795 0.748671
\(396\) −2.56397 −0.128844
\(397\) 17.9147 0.899111 0.449555 0.893252i \(-0.351582\pi\)
0.449555 + 0.893252i \(0.351582\pi\)
\(398\) −59.7776 −2.99638
\(399\) −1.53968 −0.0770806
\(400\) 3.31128 0.165564
\(401\) 16.0895 0.803473 0.401736 0.915755i \(-0.368407\pi\)
0.401736 + 0.915755i \(0.368407\pi\)
\(402\) 16.7616 0.835994
\(403\) 43.3553 2.15968
\(404\) 56.2061 2.79636
\(405\) 16.4133 0.815586
\(406\) 1.12703 0.0559338
\(407\) 5.33079 0.264237
\(408\) 6.07787 0.300900
\(409\) 19.7873 0.978421 0.489211 0.872166i \(-0.337285\pi\)
0.489211 + 0.872166i \(0.337285\pi\)
\(410\) −52.4824 −2.59192
\(411\) −32.3195 −1.59421
\(412\) −11.2120 −0.552378
\(413\) −4.11039 −0.202259
\(414\) −7.63588 −0.375283
\(415\) 24.8235 1.21854
\(416\) −14.8087 −0.726057
\(417\) 13.1263 0.642799
\(418\) −2.43379 −0.119041
\(419\) 13.4321 0.656199 0.328099 0.944643i \(-0.393592\pi\)
0.328099 + 0.944643i \(0.393592\pi\)
\(420\) 14.4725 0.706184
\(421\) 12.9939 0.633284 0.316642 0.948545i \(-0.397445\pi\)
0.316642 + 0.948545i \(0.397445\pi\)
\(422\) 2.39399 0.116538
\(423\) −4.26481 −0.207362
\(424\) −22.9149 −1.11285
\(425\) −1.29511 −0.0628219
\(426\) −50.6739 −2.45516
\(427\) −0.507136 −0.0245420
\(428\) 33.2282 1.60615
\(429\) −9.55902 −0.461514
\(430\) 32.4958 1.56709
\(431\) 31.7065 1.52725 0.763623 0.645662i \(-0.223419\pi\)
0.763623 + 0.645662i \(0.223419\pi\)
\(432\) −13.7828 −0.663126
\(433\) 2.80020 0.134569 0.0672845 0.997734i \(-0.478566\pi\)
0.0672845 + 0.997734i \(0.478566\pi\)
\(434\) −16.9959 −0.815830
\(435\) −1.79016 −0.0858318
\(436\) 33.8741 1.62228
\(437\) −4.71882 −0.225732
\(438\) 36.8758 1.76199
\(439\) −3.34540 −0.159667 −0.0798337 0.996808i \(-0.525439\pi\)
−0.0798337 + 0.996808i \(0.525439\pi\)
\(440\) 10.6144 0.506020
\(441\) 4.04205 0.192478
\(442\) −14.2038 −0.675608
\(443\) 11.9357 0.567084 0.283542 0.958960i \(-0.408491\pi\)
0.283542 + 0.958960i \(0.408491\pi\)
\(444\) 29.8266 1.41551
\(445\) −11.8230 −0.560465
\(446\) 34.1307 1.61614
\(447\) −4.77512 −0.225855
\(448\) 10.7733 0.508988
\(449\) 12.3046 0.580690 0.290345 0.956922i \(-0.406230\pi\)
0.290345 + 0.956922i \(0.406230\pi\)
\(450\) 2.17859 0.102700
\(451\) −8.84688 −0.416584
\(452\) 0.916329 0.0431005
\(453\) −18.5481 −0.871468
\(454\) 29.7578 1.39660
\(455\) −15.6927 −0.735683
\(456\) −6.31825 −0.295879
\(457\) 15.7677 0.737583 0.368791 0.929512i \(-0.379772\pi\)
0.368791 + 0.929512i \(0.379772\pi\)
\(458\) −52.8473 −2.46939
\(459\) 5.39072 0.251617
\(460\) 44.3552 2.06807
\(461\) 13.0230 0.606543 0.303271 0.952904i \(-0.401921\pi\)
0.303271 + 0.952904i \(0.401921\pi\)
\(462\) 3.74728 0.174339
\(463\) 8.30575 0.386001 0.193000 0.981199i \(-0.438178\pi\)
0.193000 + 0.981199i \(0.438178\pi\)
\(464\) 1.14644 0.0532224
\(465\) 26.9960 1.25191
\(466\) −30.3630 −1.40654
\(467\) −23.5531 −1.08991 −0.544953 0.838466i \(-0.683453\pi\)
−0.544953 + 0.838466i \(0.683453\pi\)
\(468\) 15.5554 0.719046
\(469\) 4.63848 0.214185
\(470\) 38.0523 1.75522
\(471\) 23.4204 1.07915
\(472\) −16.8674 −0.776386
\(473\) 5.47776 0.251868
\(474\) 21.5564 0.990118
\(475\) 1.34633 0.0617737
\(476\) 3.62503 0.166153
\(477\) −3.73722 −0.171116
\(478\) 5.28160 0.241575
\(479\) −26.5839 −1.21465 −0.607324 0.794454i \(-0.707757\pi\)
−0.607324 + 0.794454i \(0.707757\pi\)
\(480\) −9.22093 −0.420876
\(481\) −32.3414 −1.47464
\(482\) −35.1118 −1.59930
\(483\) 7.26550 0.330592
\(484\) −37.1870 −1.69032
\(485\) 15.9202 0.722900
\(486\) −16.4688 −0.747040
\(487\) 15.2263 0.689967 0.344984 0.938609i \(-0.387884\pi\)
0.344984 + 0.938609i \(0.387884\pi\)
\(488\) −2.08108 −0.0942063
\(489\) 12.4668 0.563768
\(490\) −36.0648 −1.62924
\(491\) 21.5944 0.974543 0.487271 0.873251i \(-0.337992\pi\)
0.487271 + 0.873251i \(0.337992\pi\)
\(492\) −49.4998 −2.23162
\(493\) −0.448396 −0.0201948
\(494\) 14.7656 0.664336
\(495\) 1.73111 0.0778076
\(496\) −17.2886 −0.776282
\(497\) −14.0231 −0.629021
\(498\) 35.9625 1.61152
\(499\) 16.5485 0.740810 0.370405 0.928870i \(-0.379219\pi\)
0.370405 + 0.928870i \(0.379219\pi\)
\(500\) 34.3432 1.53587
\(501\) 27.8925 1.24614
\(502\) −14.1497 −0.631532
\(503\) 10.6404 0.474432 0.237216 0.971457i \(-0.423765\pi\)
0.237216 + 0.971457i \(0.423765\pi\)
\(504\) −2.82932 −0.126028
\(505\) −37.9485 −1.68869
\(506\) 11.4846 0.510555
\(507\) 38.1753 1.69542
\(508\) 13.9755 0.620063
\(509\) 8.97296 0.397720 0.198860 0.980028i \(-0.436276\pi\)
0.198860 + 0.980028i \(0.436276\pi\)
\(510\) −8.84430 −0.391632
\(511\) 10.2047 0.451430
\(512\) 26.2920 1.16195
\(513\) −5.60392 −0.247419
\(514\) 38.6989 1.70694
\(515\) 7.57001 0.333575
\(516\) 30.6490 1.34925
\(517\) 6.41443 0.282106
\(518\) 12.6783 0.557052
\(519\) −22.2676 −0.977438
\(520\) −64.3964 −2.82397
\(521\) 13.6795 0.599311 0.299655 0.954048i \(-0.403128\pi\)
0.299655 + 0.954048i \(0.403128\pi\)
\(522\) 0.754280 0.0330139
\(523\) −5.19636 −0.227221 −0.113611 0.993525i \(-0.536242\pi\)
−0.113611 + 0.993525i \(0.536242\pi\)
\(524\) 62.9993 2.75214
\(525\) −2.07292 −0.0904696
\(526\) 62.4055 2.72101
\(527\) 6.76191 0.294553
\(528\) 3.81181 0.165888
\(529\) −0.732711 −0.0318570
\(530\) 33.3450 1.44841
\(531\) −2.75092 −0.119380
\(532\) −3.76840 −0.163381
\(533\) 53.6732 2.32484
\(534\) −17.1283 −0.741215
\(535\) −22.4346 −0.969934
\(536\) 19.0345 0.822164
\(537\) −1.22368 −0.0528056
\(538\) 36.9512 1.59308
\(539\) −6.07938 −0.261858
\(540\) 52.6748 2.26676
\(541\) 5.54278 0.238303 0.119151 0.992876i \(-0.461983\pi\)
0.119151 + 0.992876i \(0.461983\pi\)
\(542\) −12.0383 −0.517087
\(543\) 2.68470 0.115212
\(544\) −2.30964 −0.0990250
\(545\) −22.8707 −0.979673
\(546\) −22.7344 −0.972941
\(547\) 30.2873 1.29499 0.647496 0.762069i \(-0.275816\pi\)
0.647496 + 0.762069i \(0.275816\pi\)
\(548\) −79.1025 −3.37909
\(549\) −0.339406 −0.0144855
\(550\) −3.27668 −0.139718
\(551\) 0.466130 0.0198578
\(552\) 29.8147 1.26900
\(553\) 5.96535 0.253672
\(554\) 13.1735 0.559690
\(555\) −20.1380 −0.854810
\(556\) 32.1269 1.36248
\(557\) 23.5539 0.998011 0.499005 0.866599i \(-0.333699\pi\)
0.499005 + 0.866599i \(0.333699\pi\)
\(558\) −11.3747 −0.481529
\(559\) −33.2331 −1.40561
\(560\) 6.25770 0.264436
\(561\) −1.49087 −0.0629446
\(562\) −72.9974 −3.07921
\(563\) 0.418547 0.0176397 0.00881983 0.999961i \(-0.497193\pi\)
0.00881983 + 0.999961i \(0.497193\pi\)
\(564\) 35.8898 1.51123
\(565\) −0.618676 −0.0260279
\(566\) 70.1403 2.94822
\(567\) 6.58026 0.276345
\(568\) −57.5451 −2.41454
\(569\) 18.3361 0.768688 0.384344 0.923190i \(-0.374428\pi\)
0.384344 + 0.923190i \(0.374428\pi\)
\(570\) 9.19409 0.385098
\(571\) −6.40194 −0.267913 −0.133956 0.990987i \(-0.542768\pi\)
−0.133956 + 0.990987i \(0.542768\pi\)
\(572\) −23.3958 −0.978228
\(573\) −26.5695 −1.10996
\(574\) −21.0407 −0.878221
\(575\) −6.35308 −0.264942
\(576\) 7.21011 0.300421
\(577\) −5.67089 −0.236082 −0.118041 0.993009i \(-0.537661\pi\)
−0.118041 + 0.993009i \(0.537661\pi\)
\(578\) 38.4826 1.60067
\(579\) −22.4862 −0.934493
\(580\) −4.38145 −0.181930
\(581\) 9.95199 0.412878
\(582\) 23.0641 0.956036
\(583\) 5.62092 0.232795
\(584\) 41.8761 1.73284
\(585\) −10.5025 −0.434224
\(586\) −74.9982 −3.09815
\(587\) −39.6616 −1.63701 −0.818504 0.574501i \(-0.805196\pi\)
−0.818504 + 0.574501i \(0.805196\pi\)
\(588\) −34.0152 −1.40276
\(589\) −7.02934 −0.289639
\(590\) 24.5449 1.01050
\(591\) 25.2769 1.03975
\(592\) 12.8966 0.530049
\(593\) 29.2391 1.20071 0.600353 0.799735i \(-0.295027\pi\)
0.600353 + 0.799735i \(0.295027\pi\)
\(594\) 13.6388 0.559606
\(595\) −2.44750 −0.100338
\(596\) −11.6872 −0.478725
\(597\) 38.0662 1.55795
\(598\) −69.6762 −2.84927
\(599\) −0.362610 −0.0148158 −0.00740792 0.999973i \(-0.502358\pi\)
−0.00740792 + 0.999973i \(0.502358\pi\)
\(600\) −8.50643 −0.347273
\(601\) 19.7417 0.805282 0.402641 0.915358i \(-0.368092\pi\)
0.402641 + 0.915358i \(0.368092\pi\)
\(602\) 13.0279 0.530976
\(603\) 3.10435 0.126419
\(604\) −45.3968 −1.84717
\(605\) 25.1075 1.02076
\(606\) −54.9770 −2.23329
\(607\) −10.6293 −0.431432 −0.215716 0.976456i \(-0.569209\pi\)
−0.215716 + 0.976456i \(0.569209\pi\)
\(608\) 2.40098 0.0973728
\(609\) −0.717693 −0.0290824
\(610\) 3.02832 0.122613
\(611\) −38.9157 −1.57436
\(612\) 2.42609 0.0980688
\(613\) −14.9268 −0.602889 −0.301444 0.953484i \(-0.597469\pi\)
−0.301444 + 0.953484i \(0.597469\pi\)
\(614\) 45.3306 1.82939
\(615\) 33.4207 1.34765
\(616\) 4.25540 0.171455
\(617\) −25.2249 −1.01552 −0.507759 0.861499i \(-0.669526\pi\)
−0.507759 + 0.861499i \(0.669526\pi\)
\(618\) 10.9669 0.441152
\(619\) −36.3429 −1.46074 −0.730372 0.683049i \(-0.760653\pi\)
−0.730372 + 0.683049i \(0.760653\pi\)
\(620\) 66.0732 2.65356
\(621\) 26.4439 1.06116
\(622\) 59.2470 2.37559
\(623\) −4.73996 −0.189902
\(624\) −23.1259 −0.925777
\(625\) −29.9190 −1.19676
\(626\) −12.0366 −0.481078
\(627\) 1.54983 0.0618944
\(628\) 57.3216 2.28738
\(629\) −5.04412 −0.201122
\(630\) 4.11712 0.164030
\(631\) 8.68733 0.345837 0.172919 0.984936i \(-0.444680\pi\)
0.172919 + 0.984936i \(0.444680\pi\)
\(632\) 24.4794 0.973738
\(633\) −1.52449 −0.0605930
\(634\) 26.7618 1.06285
\(635\) −9.43582 −0.374449
\(636\) 31.4500 1.24707
\(637\) 36.8830 1.46136
\(638\) −1.13446 −0.0449139
\(639\) −9.38510 −0.371269
\(640\) −52.2344 −2.06475
\(641\) −5.41728 −0.213970 −0.106985 0.994261i \(-0.534120\pi\)
−0.106985 + 0.994261i \(0.534120\pi\)
\(642\) −32.5017 −1.28274
\(643\) −5.62772 −0.221936 −0.110968 0.993824i \(-0.535395\pi\)
−0.110968 + 0.993824i \(0.535395\pi\)
\(644\) 17.7824 0.700725
\(645\) −20.6932 −0.814795
\(646\) 2.30292 0.0906070
\(647\) −46.6460 −1.83384 −0.916922 0.399066i \(-0.869334\pi\)
−0.916922 + 0.399066i \(0.869334\pi\)
\(648\) 27.0028 1.06077
\(649\) 4.13749 0.162411
\(650\) 19.8793 0.779731
\(651\) 10.8230 0.424185
\(652\) 30.5126 1.19497
\(653\) 42.2253 1.65240 0.826202 0.563374i \(-0.190497\pi\)
0.826202 + 0.563374i \(0.190497\pi\)
\(654\) −33.1334 −1.29562
\(655\) −42.5351 −1.66198
\(656\) −21.4030 −0.835649
\(657\) 6.82961 0.266449
\(658\) 15.2555 0.594722
\(659\) 27.0148 1.05235 0.526173 0.850378i \(-0.323626\pi\)
0.526173 + 0.850378i \(0.323626\pi\)
\(660\) −14.5679 −0.567053
\(661\) 41.5902 1.61767 0.808836 0.588034i \(-0.200098\pi\)
0.808836 + 0.588034i \(0.200098\pi\)
\(662\) 56.1529 2.18244
\(663\) 9.04497 0.351278
\(664\) 40.8390 1.58486
\(665\) 2.54430 0.0986637
\(666\) 8.48508 0.328790
\(667\) −2.19959 −0.0851683
\(668\) 68.2671 2.64133
\(669\) −21.7343 −0.840298
\(670\) −27.6983 −1.07008
\(671\) 0.510480 0.0197068
\(672\) −3.69676 −0.142606
\(673\) −42.6309 −1.64330 −0.821649 0.569993i \(-0.806946\pi\)
−0.821649 + 0.569993i \(0.806946\pi\)
\(674\) 61.2343 2.35866
\(675\) −7.54470 −0.290396
\(676\) 93.4345 3.59363
\(677\) 3.45734 0.132876 0.0664382 0.997791i \(-0.478836\pi\)
0.0664382 + 0.997791i \(0.478836\pi\)
\(678\) −0.896292 −0.0344219
\(679\) 6.38257 0.244940
\(680\) −10.0436 −0.385154
\(681\) −18.9497 −0.726155
\(682\) 17.1080 0.655098
\(683\) 47.2337 1.80735 0.903674 0.428222i \(-0.140860\pi\)
0.903674 + 0.428222i \(0.140860\pi\)
\(684\) −2.52204 −0.0964326
\(685\) 53.4075 2.04059
\(686\) −31.3837 −1.19823
\(687\) 33.6530 1.28394
\(688\) 13.2522 0.505236
\(689\) −34.1016 −1.29917
\(690\) −43.3853 −1.65165
\(691\) −4.24312 −0.161416 −0.0807080 0.996738i \(-0.525718\pi\)
−0.0807080 + 0.996738i \(0.525718\pi\)
\(692\) −54.5001 −2.07178
\(693\) 0.694018 0.0263636
\(694\) 36.4582 1.38393
\(695\) −21.6910 −0.822787
\(696\) −2.94513 −0.111635
\(697\) 8.37114 0.317079
\(698\) −24.8671 −0.941235
\(699\) 19.3351 0.731321
\(700\) −5.07349 −0.191760
\(701\) 13.9967 0.528648 0.264324 0.964434i \(-0.414851\pi\)
0.264324 + 0.964434i \(0.414851\pi\)
\(702\) −82.7452 −3.12302
\(703\) 5.24361 0.197767
\(704\) −10.8443 −0.408709
\(705\) −24.2316 −0.912616
\(706\) −54.2947 −2.04341
\(707\) −15.2139 −0.572178
\(708\) 23.1500 0.870029
\(709\) 29.4882 1.10745 0.553726 0.832699i \(-0.313206\pi\)
0.553726 + 0.832699i \(0.313206\pi\)
\(710\) 83.7376 3.14262
\(711\) 3.99237 0.149726
\(712\) −19.4509 −0.728953
\(713\) 33.1702 1.24223
\(714\) −3.54576 −0.132697
\(715\) 15.7961 0.590741
\(716\) −2.99497 −0.111927
\(717\) −3.36331 −0.125605
\(718\) −64.4451 −2.40507
\(719\) 8.96886 0.334482 0.167241 0.985916i \(-0.446514\pi\)
0.167241 + 0.985916i \(0.446514\pi\)
\(720\) 4.18803 0.156079
\(721\) 3.03489 0.113025
\(722\) −2.39399 −0.0890952
\(723\) 22.3591 0.831544
\(724\) 6.57085 0.244204
\(725\) 0.627563 0.0233071
\(726\) 36.3738 1.34996
\(727\) −21.0906 −0.782207 −0.391104 0.920347i \(-0.627907\pi\)
−0.391104 + 0.920347i \(0.627907\pi\)
\(728\) −25.8171 −0.956845
\(729\) 30.0333 1.11234
\(730\) −60.9365 −2.25536
\(731\) −5.18319 −0.191707
\(732\) 2.85622 0.105569
\(733\) −26.2316 −0.968887 −0.484443 0.874823i \(-0.660978\pi\)
−0.484443 + 0.874823i \(0.660978\pi\)
\(734\) 55.3089 2.04149
\(735\) 22.9660 0.847112
\(736\) −11.3298 −0.417623
\(737\) −4.66906 −0.171987
\(738\) −14.0817 −0.518354
\(739\) −26.4720 −0.973789 −0.486894 0.873461i \(-0.661870\pi\)
−0.486894 + 0.873461i \(0.661870\pi\)
\(740\) −49.2880 −1.81186
\(741\) −9.40270 −0.345417
\(742\) 13.3683 0.490767
\(743\) −17.0643 −0.626030 −0.313015 0.949748i \(-0.601339\pi\)
−0.313015 + 0.949748i \(0.601339\pi\)
\(744\) 44.4131 1.62826
\(745\) 7.89080 0.289096
\(746\) −42.2434 −1.54664
\(747\) 6.66047 0.243694
\(748\) −3.64893 −0.133418
\(749\) −8.99426 −0.328643
\(750\) −33.5922 −1.22661
\(751\) 39.5414 1.44289 0.721443 0.692474i \(-0.243479\pi\)
0.721443 + 0.692474i \(0.243479\pi\)
\(752\) 15.5183 0.565893
\(753\) 9.01049 0.328360
\(754\) 6.88269 0.250653
\(755\) 30.6505 1.11548
\(756\) 21.1178 0.768047
\(757\) −29.0014 −1.05407 −0.527037 0.849842i \(-0.676697\pi\)
−0.527037 + 0.849842i \(0.676697\pi\)
\(758\) −8.70287 −0.316102
\(759\) −7.31340 −0.265459
\(760\) 10.4408 0.378727
\(761\) −41.3892 −1.50036 −0.750180 0.661234i \(-0.770033\pi\)
−0.750180 + 0.661234i \(0.770033\pi\)
\(762\) −13.6699 −0.495209
\(763\) −9.16908 −0.331943
\(764\) −65.0291 −2.35267
\(765\) −1.63802 −0.0592226
\(766\) −40.8018 −1.47423
\(767\) −25.1018 −0.906372
\(768\) −43.1502 −1.55705
\(769\) 15.0461 0.542578 0.271289 0.962498i \(-0.412550\pi\)
0.271289 + 0.962498i \(0.412550\pi\)
\(770\) −6.19231 −0.223155
\(771\) −24.6434 −0.887510
\(772\) −55.0352 −1.98076
\(773\) 32.0186 1.15163 0.575814 0.817581i \(-0.304685\pi\)
0.575814 + 0.817581i \(0.304685\pi\)
\(774\) 8.71903 0.313399
\(775\) −9.46378 −0.339949
\(776\) 26.1915 0.940220
\(777\) −8.07351 −0.289635
\(778\) −21.3777 −0.766429
\(779\) −8.70221 −0.311789
\(780\) 88.3819 3.16458
\(781\) 14.1155 0.505093
\(782\) −10.8670 −0.388605
\(783\) −2.61216 −0.0933508
\(784\) −14.7077 −0.525275
\(785\) −38.7017 −1.38132
\(786\) −61.6217 −2.19797
\(787\) 13.8389 0.493302 0.246651 0.969104i \(-0.420670\pi\)
0.246651 + 0.969104i \(0.420670\pi\)
\(788\) 61.8656 2.20387
\(789\) −39.7397 −1.41477
\(790\) −35.6215 −1.26736
\(791\) −0.248033 −0.00881903
\(792\) 2.84797 0.101198
\(793\) −3.09703 −0.109979
\(794\) −42.8876 −1.52202
\(795\) −21.2340 −0.753093
\(796\) 93.1676 3.30224
\(797\) −23.7849 −0.842505 −0.421252 0.906943i \(-0.638409\pi\)
−0.421252 + 0.906943i \(0.638409\pi\)
\(798\) 3.68600 0.130483
\(799\) −6.06948 −0.214723
\(800\) 3.23251 0.114286
\(801\) −3.17227 −0.112086
\(802\) −38.5182 −1.36013
\(803\) −10.2720 −0.362491
\(804\) −26.1242 −0.921329
\(805\) −12.0061 −0.423160
\(806\) −103.792 −3.65593
\(807\) −23.5304 −0.828310
\(808\) −62.4318 −2.19634
\(809\) −53.1637 −1.86914 −0.934568 0.355784i \(-0.884214\pi\)
−0.934568 + 0.355784i \(0.884214\pi\)
\(810\) −39.2935 −1.38063
\(811\) −24.3701 −0.855750 −0.427875 0.903838i \(-0.640738\pi\)
−0.427875 + 0.903838i \(0.640738\pi\)
\(812\) −1.75656 −0.0616433
\(813\) 7.66593 0.268856
\(814\) −12.7619 −0.447303
\(815\) −20.6011 −0.721627
\(816\) −3.60683 −0.126264
\(817\) 5.38819 0.188509
\(818\) −47.3708 −1.65628
\(819\) −4.21054 −0.147128
\(820\) 81.7975 2.85649
\(821\) −32.4003 −1.13078 −0.565389 0.824824i \(-0.691274\pi\)
−0.565389 + 0.824824i \(0.691274\pi\)
\(822\) 77.3728 2.69869
\(823\) 54.5118 1.90016 0.950082 0.312001i \(-0.100999\pi\)
0.950082 + 0.312001i \(0.100999\pi\)
\(824\) 12.4540 0.433854
\(825\) 2.08658 0.0726455
\(826\) 9.84026 0.342386
\(827\) 37.0810 1.28943 0.644716 0.764422i \(-0.276975\pi\)
0.644716 + 0.764422i \(0.276975\pi\)
\(828\) 11.9011 0.413590
\(829\) −18.0895 −0.628274 −0.314137 0.949378i \(-0.601715\pi\)
−0.314137 + 0.949378i \(0.601715\pi\)
\(830\) −59.4274 −2.06276
\(831\) −8.38888 −0.291007
\(832\) 65.7912 2.28090
\(833\) 5.75246 0.199311
\(834\) −31.4244 −1.08814
\(835\) −46.0918 −1.59507
\(836\) 3.79324 0.131192
\(837\) 39.3918 1.36158
\(838\) −32.1563 −1.11082
\(839\) −30.5902 −1.05609 −0.528046 0.849216i \(-0.677075\pi\)
−0.528046 + 0.849216i \(0.677075\pi\)
\(840\) −16.0755 −0.554658
\(841\) −28.7827 −0.992508
\(842\) −31.1073 −1.07203
\(843\) 46.4846 1.60101
\(844\) −3.73121 −0.128433
\(845\) −63.0840 −2.17015
\(846\) 10.2099 0.351024
\(847\) 10.0658 0.345865
\(848\) 13.5986 0.466976
\(849\) −44.6652 −1.53290
\(850\) 3.10048 0.106345
\(851\) −24.7437 −0.848202
\(852\) 78.9788 2.70577
\(853\) −12.5604 −0.430061 −0.215030 0.976607i \(-0.568985\pi\)
−0.215030 + 0.976607i \(0.568985\pi\)
\(854\) 1.21408 0.0415450
\(855\) 1.70280 0.0582345
\(856\) −36.9088 −1.26152
\(857\) −13.5791 −0.463852 −0.231926 0.972733i \(-0.574503\pi\)
−0.231926 + 0.972733i \(0.574503\pi\)
\(858\) 22.8842 0.781255
\(859\) 1.04768 0.0357463 0.0178731 0.999840i \(-0.494311\pi\)
0.0178731 + 0.999840i \(0.494311\pi\)
\(860\) −50.6469 −1.72705
\(861\) 13.3987 0.456625
\(862\) −75.9051 −2.58534
\(863\) 32.8144 1.11701 0.558507 0.829500i \(-0.311374\pi\)
0.558507 + 0.829500i \(0.311374\pi\)
\(864\) −13.4549 −0.457746
\(865\) 36.7967 1.25113
\(866\) −6.70366 −0.227800
\(867\) −24.5056 −0.832255
\(868\) 26.4893 0.899107
\(869\) −6.00467 −0.203695
\(870\) 4.28564 0.145297
\(871\) 28.3267 0.959815
\(872\) −37.6262 −1.27418
\(873\) 4.27160 0.144572
\(874\) 11.2968 0.382121
\(875\) −9.29604 −0.314264
\(876\) −57.4735 −1.94185
\(877\) −12.5291 −0.423078 −0.211539 0.977370i \(-0.567848\pi\)
−0.211539 + 0.977370i \(0.567848\pi\)
\(878\) 8.00888 0.270287
\(879\) 47.7587 1.61086
\(880\) −6.29895 −0.212338
\(881\) −44.9832 −1.51552 −0.757761 0.652532i \(-0.773707\pi\)
−0.757761 + 0.652532i \(0.773707\pi\)
\(882\) −9.67663 −0.325829
\(883\) 23.2820 0.783502 0.391751 0.920071i \(-0.371869\pi\)
0.391751 + 0.920071i \(0.371869\pi\)
\(884\) 22.1377 0.744571
\(885\) −15.6301 −0.525400
\(886\) −28.5741 −0.959965
\(887\) 11.0362 0.370559 0.185280 0.982686i \(-0.440681\pi\)
0.185280 + 0.982686i \(0.440681\pi\)
\(888\) −33.1304 −1.11178
\(889\) −3.78291 −0.126875
\(890\) 28.3042 0.948760
\(891\) −6.62364 −0.221900
\(892\) −53.1951 −1.78110
\(893\) 6.30953 0.211140
\(894\) 11.4316 0.382330
\(895\) 2.02210 0.0675915
\(896\) −20.9413 −0.699599
\(897\) 44.3697 1.48146
\(898\) −29.4571 −0.982997
\(899\) −3.27659 −0.109280
\(900\) −3.39549 −0.113183
\(901\) −5.31865 −0.177190
\(902\) 21.1794 0.705197
\(903\) −8.29611 −0.276077
\(904\) −1.01783 −0.0338524
\(905\) −4.43642 −0.147472
\(906\) 44.4041 1.47523
\(907\) −9.64206 −0.320159 −0.160080 0.987104i \(-0.551175\pi\)
−0.160080 + 0.987104i \(0.551175\pi\)
\(908\) −46.3797 −1.53916
\(909\) −10.1821 −0.337718
\(910\) 37.5681 1.24537
\(911\) 14.0619 0.465893 0.232946 0.972490i \(-0.425163\pi\)
0.232946 + 0.972490i \(0.425163\pi\)
\(912\) 3.74948 0.124158
\(913\) −10.0176 −0.331534
\(914\) −37.7478 −1.24859
\(915\) −1.92843 −0.0637518
\(916\) 82.3662 2.72146
\(917\) −17.0527 −0.563130
\(918\) −12.9054 −0.425940
\(919\) −2.73360 −0.0901730 −0.0450865 0.998983i \(-0.514356\pi\)
−0.0450865 + 0.998983i \(0.514356\pi\)
\(920\) −49.2683 −1.62433
\(921\) −28.8664 −0.951181
\(922\) −31.1771 −1.02676
\(923\) −85.6376 −2.81880
\(924\) −5.84039 −0.192135
\(925\) 7.05961 0.232119
\(926\) −19.8839 −0.653426
\(927\) 2.03113 0.0667111
\(928\) 1.11917 0.0367386
\(929\) −48.4431 −1.58937 −0.794684 0.607024i \(-0.792363\pi\)
−0.794684 + 0.607024i \(0.792363\pi\)
\(930\) −64.6284 −2.11925
\(931\) −5.97997 −0.195985
\(932\) 47.3229 1.55011
\(933\) −37.7283 −1.23517
\(934\) 56.3859 1.84500
\(935\) 2.46364 0.0805696
\(936\) −17.2784 −0.564761
\(937\) 8.45475 0.276205 0.138102 0.990418i \(-0.455900\pi\)
0.138102 + 0.990418i \(0.455900\pi\)
\(938\) −11.1045 −0.362575
\(939\) 7.66486 0.250133
\(940\) −59.3072 −1.93439
\(941\) −23.8122 −0.776257 −0.388128 0.921605i \(-0.626878\pi\)
−0.388128 + 0.921605i \(0.626878\pi\)
\(942\) −56.0682 −1.82680
\(943\) 41.0642 1.33723
\(944\) 10.0097 0.325789
\(945\) −14.2581 −0.463815
\(946\) −13.1137 −0.426364
\(947\) −54.0395 −1.75605 −0.878023 0.478618i \(-0.841138\pi\)
−0.878023 + 0.478618i \(0.841138\pi\)
\(948\) −33.5971 −1.09118
\(949\) 62.3192 2.02297
\(950\) −3.22310 −0.104571
\(951\) −17.0419 −0.552620
\(952\) −4.02656 −0.130502
\(953\) −41.5711 −1.34662 −0.673310 0.739361i \(-0.735128\pi\)
−0.673310 + 0.739361i \(0.735128\pi\)
\(954\) 8.94689 0.289666
\(955\) 43.9056 1.42075
\(956\) −8.23175 −0.266234
\(957\) 0.722424 0.0233527
\(958\) 63.6416 2.05617
\(959\) 21.4115 0.691415
\(960\) 40.9662 1.32218
\(961\) 18.4116 0.593922
\(962\) 77.4251 2.49628
\(963\) −6.01950 −0.193976
\(964\) 54.7242 1.76255
\(965\) 37.1580 1.19616
\(966\) −17.3936 −0.559629
\(967\) −13.8825 −0.446430 −0.223215 0.974769i \(-0.571655\pi\)
−0.223215 + 0.974769i \(0.571655\pi\)
\(968\) 41.3061 1.32763
\(969\) −1.46649 −0.0471105
\(970\) −38.1129 −1.22373
\(971\) −40.8260 −1.31017 −0.655085 0.755556i \(-0.727367\pi\)
−0.655085 + 0.755556i \(0.727367\pi\)
\(972\) 25.6678 0.823294
\(973\) −8.69613 −0.278785
\(974\) −36.4516 −1.16798
\(975\) −12.6591 −0.405416
\(976\) 1.23499 0.0395311
\(977\) 32.6035 1.04308 0.521540 0.853227i \(-0.325358\pi\)
0.521540 + 0.853227i \(0.325358\pi\)
\(978\) −29.8454 −0.954351
\(979\) 4.77120 0.152488
\(980\) 56.2095 1.79555
\(981\) −6.13650 −0.195923
\(982\) −51.6969 −1.64972
\(983\) 49.1997 1.56923 0.784614 0.619984i \(-0.212861\pi\)
0.784614 + 0.619984i \(0.212861\pi\)
\(984\) 54.9827 1.75279
\(985\) −41.7697 −1.33089
\(986\) 1.07346 0.0341859
\(987\) −9.71468 −0.309222
\(988\) −23.0132 −0.732148
\(989\) −25.4259 −0.808496
\(990\) −4.14427 −0.131713
\(991\) −45.9955 −1.46110 −0.730548 0.682862i \(-0.760735\pi\)
−0.730548 + 0.682862i \(0.760735\pi\)
\(992\) −16.8773 −0.535856
\(993\) −35.7580 −1.13475
\(994\) 33.5712 1.06481
\(995\) −62.9038 −1.99418
\(996\) −56.0501 −1.77602
\(997\) −42.4831 −1.34545 −0.672727 0.739891i \(-0.734877\pi\)
−0.672727 + 0.739891i \(0.734877\pi\)
\(998\) −39.6169 −1.25405
\(999\) −29.3848 −0.929693
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 4009.2.a.e.1.6 82
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4009.2.a.e.1.6 82 1.1 even 1 trivial